Research Article | Open Access

Mehmet Gümüş, Özkan Öcalan, Nilüfer B. Felah, "On the Dynamics of the Recursive Sequence ", *Discrete Dynamics in Nature and Society*, vol. 2012, Article ID 241303, 11 pages, 2012. https://doi.org/10.1155/2012/241303

# On the Dynamics of the Recursive Sequence

**Academic Editor:**M. De la Sen

#### Abstract

We investigate the boundedness character, the oscillatory, and the periodic character of positive solutions of the difference equation , where , , and the initial conditions are arbitrary positive numbers. We investigate the boundedness character for . Also, we investigate the existence of a prime two periodic solution for is odd. Moreover, when is even, we prove that there are no prime two periodic solutions of the equation above.

#### 1. Introduction

Our aim in this paper is to study the boundedness character, the oscillatory, and the periodic character of positive solutions of the difference equation where , is a positive, and the initial conditions are arbitrary positive numbers. Equation (1.1) was studied by many authors for different cases of .

In [1] the authors studied the global stability, the boundedness character, and the periodic nature of the positive solutions of the difference equation where is positive, and the initial values are positive numbers (see also [2–4] for more results on this equation).

In [5] the authors studied the boundedness, the global attractivity, the oscillatory behaviour, and the periodicity of the positive solutions of the difference equation where , are positive, and the initial values are positive numbers (see also [6–8] for more results on this equation).

In [9] the authors studied general properties, the boundedness, the global stability, and the periodic character of the solutions of the difference equation where are positive, , and the initial values are positive numbers.

In [10, 11] the authors studied the boundedness, the persistence, the attractivity, the stability, and the periodic character of the positive solutions of the difference equation where , , are positive, and the initial values , are positive numbers.

Finally in [12, 13] the authors studied the oscillatory, the behaviour of semicycle, and the periodic character of the positive solution of the difference equation where and , under the initial conditions are positive numbers.

There exist many other papers related with (1.1) and on its extensions (see [14–16]).

Motivated by the above papers, we study of the boundedness character, the oscillatory, and the periodic character of positive solutions of (1.1).

In this paper, also we investigate the case , and of (1.1) and we give a correction for [2, Theorem 2.5].

We say that the equilibrium point of the equation is the point that satisfies the condition A solution of (1.1) is called nonoscillatory if there exists such that either or A solution of (1.1) is called oscillatory if it is not nonoscillatory. We say that a solution of (1.1) is bounded and persists if there exist positive constant and such that for .

The linearized equation for (1.1) about the positive equilibrium is

#### 2. Semicycle Analysis

A positive semicycle of a solution of (1.1) consists of a “string” of terms all greater than or equal to , with and , such that A negative semicycle of a solution of (1.1) consists of a “string” of terms all less than , with and , such that

Lemma 2.1. * Let be a solution of (1.1). Then either consists of a single semicycle or oscillates about equilibrium with semicycles having at most terms.*

*Proof. *Suppose that has at least two semicycles. Then there exists such that either or . We assume that the former case holds. The latter case is similar and will be omitted. Now suppose that the positive semicycle beginning with the term has terms. Then and so the case
holds for every , , from which the result follows.

#### 3. Boundedness and Global Stability of (1.1)

In this section, we consider the case with no restriction on other parameters and we consider the case with some specified conditions. For these cases, we have the following results which give a complete picture as regards the boundedness character of positive solutions of (1.1).

Theorem 3.1. * Suppose that
**
then every positive solution of (1.1) is bounded.*

*Proof. *We have
Hence we will prove that is bounded. If
then we have , . Hence, the function is increasing and concave. Thus, we get that there is a unique fixed point of the equation . Also the function satisfies the condition
Using [15, 2.6.2] we obtain that is a global attractor of all positive solutions of (1.1) and so is bounded, from which the result follows.

Now we study the boundedness of (1.1) for the case . We give better result than Theorem 3.1 for the boundedness of (1.1) and we prove that in this case, there exist unbounded solutions of (1.1).

Theorem 3.2. * Consider (1.1) and assume that , , and . Then every positive solution of (1.1) is bounded and . *

*Proof. *Let
Suppose on the contrary that every positive solution of (1.1) is unbounded. Then, from (1.1), we obtain for . Therefore we get
Thus the proof is complete. We note that in here is a continuous function for .

Theorem 3.3. * Consider (1.1) when the case is odd and suppose that
**
then there exists unbounded solutions of (1.1).*

*Proof. *Let be a solution of (1.1) with initial values such that
Then from (1.1), (3.7), and (3.8) we have
Also, from (3.8) and (3.10)–(3.14), it is clear that
Moreover from (1.1) and (3.8)–(3.14) and arguing as above we get
Therefore working inductively we can prove that for ,
which implies that
So is unbounded. From which the result follows.

Now, in the next theorem, we will provide an alternative proof for the theorem above when and is odd, whose proof can be used for some practical applications.

Theorem 3.4. * Consider (1.1) when the case is odd and suppose that . If , then there exists solutions of (1.1) that are unbounded.*

*Proof. *We assume that and choose the initial conditions such that
So,
Therefore, we obtain and . By induction, for , we have and . Thus,
Now, we assume that and choose the initial conditions such that
So, we have
Further, we have
By induction for , we have and . Thus,
from which the result follows.

The following result is essentially proved in [10, 11] for . The result is satisfied for and its proof is omitted.

Lemma 3.5. * If Either
**
or
**
holds, then (1.1) has a unique equilibrium point . *

The following result is essentially proved in [10, 11] for . It is clear that the result is satisfied when is odd and its proof is omitted.

Lemma 3.6. * Consider (1.1) when the case is odd. Suppose that
**
or
**
holds. Then the unique positive equilibrium of (1.1) is globally asymptotically stable.*

#### 4. Periodicity of the Solutions of (1.1)

In this section, we investigate the existence of a prime two periodic solution for is odd. Moreover, when is even, we prove that there are no positive prime two periodic solutions and lastly, we give a correction for Theorem 2.5 which was given in [2].

The following result is given when the case in [10]. If is odd, the result is still satisfied and its proof is omitted.

Lemma 4.1. * Assume that k is odd. Then, (1.1) has prime two periodic solutions if and only if
**
and there exists a sufficient small positive number , such that
*

Now, let consider the case where is even.

Theorem 4.2. * Consider (1.1) when the case is even and the following conditions are satisfied separately:
**
Then, there are no positive prime two periodic solutions of (1.1).*

*Proof. *Firstly, we consider the case and is even of (1.1) and suppose that
where , is a prime two periodic solution of (1.1). Then it must be
Substituting (4.6) into (4.5), it follows that
Taking logarithm on both sides of (4.7), we obtain that
So from (4.8)
From , thus , which implies that is a unique solution of (1.1).

We consider the case and is even of (1.1). The proof of this case is similar to the first case’s proof and will be omitted.

Now, suppose that , and is even of (1.1). In this case we have that
Considering the numerator on the right hand side in (4.10) let
From and ,
which implies that is a unique solution of (1.1).

Suppose that , and is even of (1.1). The proof of this case is similar to the third case’s proof and will be omitted.

The following result was given in [2, Theorem 2.5] for (1.1) when the case , and . But the authors make some mistakes in this theorem. Now, we give a correction and a conjecture for this result.

Theorem A. *Consider (1.1). Let be , , , and . Suppose that
**
hold. Then the unique positive equilibrium of (1.1) is globally asymptotically stable.*

* Correction B*

Consider (1.1). Let be , , , and . Suppose that
holds. Then the unique positive equilibrium of (1.1) is globally asymptotically stable.

*Proof. *It is easy to see the proof from Theorem 2.5 in [2].

Conjecture 4.3. * Consider (1.1). Let be , , , and . Suppose that
holds. Then the unique positive equilibrium of (1.1) is globally asymptotically stable.*

#### References

- A. M. Amleh, E. A. Grove, G. Ladas, and D. A. Georgiou, “On the recursive sequence ${x}_{n+1}=\alpha +({x}_{n-1}/{x}_{n})$,”
*Journal of Mathematical Analysis and Applications*, vol. 233, no. 2, pp. 790–798, 1999. View at: Publisher Site | Google Scholar | Zentralblatt MATH - M. Gümüş and Ö. Öcalan, “Some notes on the difference equation ${x}_{n+1}=\alpha +({x}_{n-1}^{}/{x}_{n}^{k})$,”
*Discrete Dynamics in Nature and Society*, vol. 2012, Article ID 258502, 12 pages, 2012. View at: Publisher Site | Google Scholar - S. Stević, “On the recursive sequence ${x}_{n+1}={\alpha}_{n}+({x}_{n-1}^{}/{x}_{n}^{})$. II,”
*Dynamics of Continuous, Discrete & Impulsive Systems A*, vol. 10, no. 6, pp. 911–916, 2003. View at: Google Scholar - S. Stević, “On the recursive sequence ${x}_{n+1}={\alpha}_{n}+({x}_{n-1}^{}/{x}_{n}^{})$,”
*International Journal of Mathematical Sciences*, vol. 2, no. 2, pp. 237–243, 2003. View at: Google Scholar | Zentralblatt MATH - K. S. Berenhaut and S. Stević, “The behaviour of the positive solutions of the difference equation ${x}_{n}=A+({x}_{n-2}/{x}_{n-1})$
^{p},”*Journal of Difference Equations and Applications*, vol. 12, no. 9, pp. 909–918, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH - H. M. El-Owaidy, A. M. Ahmed, and M. S. Mousa, “On asymptotic behaviour of the difference equation ${x}_{n+1}=\alpha +({x}_{n-1}^{p}/{x}_{n}^{p})$,”
*Journal of Applied Mathematics & Computing*, vol. 12, no. 1-2, pp. 31–37, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH - S. Stević, “On the recursive sequence ${x}_{n+1}=\alpha +({x}_{n-1}^{p}/{x}_{n}^{p})$,”
*Journal of Applied Mathematics & Computing*, vol. 18, no. 1-2, pp. 229–234, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH - S. Stević, “On the recursive sequence ${x}_{n+1}=\alpha +({x}_{n}^{p}/{x}_{n-1}^{p})$,”
*Discrete Dynamics in Nature and Society*, vol. 2007, Article ID 34517, 9 pages, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH - R. DeVault, C. Kent, and W. Kosmala, “On the recursive sequence ${x}_{n+1}=p+({x}_{n-k}/{x}_{n})$,”
*Journal of Difference Equations and Applications*, vol. 9, no. 8, pp. 721–730, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH - Ö. Öcalan, “Boundedness and stability behavior of the recursive sequence ${x}_{n+1}=A+({x}_{n-k}^{p}/{x}_{n-m}^{q})$,” In press. View at: Google Scholar
- C. J. Schinas, G. Papaschinopoulos, and G. Stefanidou, “On the recursive sequence ${x}_{n+1}=A+({x}_{n-1}^{p}/{x}_{n}^{q})$,”
*Advences in Difference Equations*, vol. 2009, Article ID 327649, 11 pages, 2009. View at: Google Scholar - S. Stević, “On the recursive sequence ${x}_{n}=\text{}1+{\sum}_{i=1}^{k}{\alpha}_{i}{x}_{n-{p}_{i}}/{\sum}_{j=1}^{m}{\beta}_{j}{x}_{n-{q}_{j}}$,”
*Discrete Dynamics in Nature and Society*, vol. 2007, Article ID 39404, 7 pages, 2007. View at: Google Scholar | Zentralblatt MATH - S. Stević, “On a class of higher-order difference equations,”
*Chaos, Solitons and Fractals*, vol. 42, no. 1, pp. 138–145, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH - S. N. Elaydi,
*An Introduction to Difference Equations*, Springer, New York, NY, USA, 1996. - V. L. Kocić and G. Ladas,
*Global Behavior of Nonlinear Difference Equations of Higher Order with Applications*, vol. 256, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1993. - M. R. S. Kulenović and G. Ladas,
*Dynamics of Second Order Rational Difference Equations*, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2002.

#### Copyright

Copyright © 2012 Mehmet Gümüş et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.